We study existence and uniqueness of nonnegative solutions to a problem which is modeled by $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta _p u = u^{-\theta }|\nabla u|^p + fu^{-\gamma }& \text {in}\, \Omega , \\ u=0 & \text {on}\ \partial \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega\) is an open bounded subset of \({\mathbb {R}}^N\) (\(N\ge 2\)), \(\Delta _p\) is

In this work, we establish the existence of nonzero solutions for a class of quasilinear elliptic equations involving indefinite nonlinearities with exponential critical growth of Trudinger–Moser type. Our proofs rely on variational arguments in a Orlicz–Sobolev space with a version of the Trudinger–Moser inequality.

In this paper, we study balanced metrics and Berezin quantization on a class of Hartogs domains defined by \(\varOmega _n=\{(z_1,\ldots ,z_n)\in {\mathbb {C}}^n:\vert z_1\vert<\vert z_2\vert<\cdots<\vert z_n\vert <1\}\) which generalize the so-called classical Hartogs triangle. We introduce a Kähler metric \(g(\nu )\) associated with the Kähler potential \(\varPhi _n(z):=-\sum _{k=1}^{n-1}\nu _k\ln

Motivated by the Hilbert-space model for quantum mechanics, we define a pre-Hilbert space logic to be a pair \((S,{\mathscr {L}})\), where S is a pre-Hilbert space and \({\mathscr {L}}\) is an orthocomplemented poset of orthogonally closed linear subspaces of S, closed w.r.t. finite-dimensional perturbations (i.e., if \(M\in {\mathscr {L}}\) and F is a finite-dimensional linear subspace of S, then

Fueter’s theorem (1934) asserts that every holomorphic intrinsic function of one complex variable induces an axial quaternionic monogenic function. Sce (Atti Accad Naz Lincei Rend Cl Sci Fis Mat Nat 23:220–225, 1957) generalizes Fueter’s theorem to the Euclidean spaces \({{\mathbb {R}}}^{n+1}\) for n being odd positive integers. By using pointwise differential computation he asserted that every holomorphic

The X-rank of a point p in projective space is the minimal number of points of an algebraic variety X whose linear span contains p. This notion is naturally submultiplicative under tensor product. We study geometric conditions that guarantee strict submultiplicativity. We prove that in the case of points of rank two, strict submultiplicativity is entirely characterized in terms of the trisecant lines

We construct an unbounded strictly pseudoconvex Kobayashi hyperbolic and complete domain in \({\mathbb {C}}^2\), which also possesses complete Bergman metric, but has no nonconstant bounded holomorphic functions.

This paper is a contribution to frame theory. Frames in a Hilbert space are generalizations of orthonormal bases. In particular, Gabor frames of \(L^2(\mathbb {R})\), which are made of translations and modulations of one or more windows, are often used in applications. More precisely, the paper deals with a question posed in the last years by Christensen and Hasannasab about the existence of overcomplete

This paper establishes a connection between a problem in Potential Theory and Mathematical Physics, arranging points so as to minimize an energy functional, and a problem in Combinatorics and Number Theory, constructing ’well-distributed’ sequences of points on [0, 1). Let \(f:[0,1] \rightarrow {\mathbb {R}}\) be (1) symmetric \(f(x) = f(1-x)\), (2) twice differentiable on (0, 1), and (3) such that

We assume to have information about the generating properties of the subsets of a finite group G. In particular, we consider the two following situations. We know, for every subset X of G, whether X is a generating set of G. We know the graph whose vertices are the subsets of G and in which there is an edge connecting X and Y if and only if \(X\cup Y\) is a generating set of G. We discuss how this

We extended the study of the linear fractional self maps (e.g., by Cowen–MacCluer and Bisi–Bracci on the unit balls) to a much more general class of domains, called generalized type I domains, which includes in particular the classical bounded symmetric domains of type I and the generalized balls. Since the linear fractional maps on the unit balls are simply the restrictions of the linear maps of the

We prove that if the elliptic problem \(-\Delta u+b(x)|\nabla u|=c(x)u\) with \(c\ge 0\) has a positive supersolution in a domain \(\varOmega \) of \( {\mathrm {R}}^{N\ge 3}\), then c, b must satisfy the inequality $$\begin{aligned} \sqrt{ \int _\varOmega c\phi ^2}\le \sqrt{ \int _\varOmega | \nabla \phi |^2}+\sqrt{ \int _\varOmega \frac{b^2}{4}\phi ^2},\quad \phi \in C_c^\infty (\varOmega ). \end{aligned}$$

We consider here three-dimensional water flows governed by the geophysical water wave equations exhibiting full Coriolis and centripetal terms. More precisely, assuming a constant vorticity vector, we derive a family of explicit solutions, in Eulerian coordinates, to the above-mentioned equations and their boundary conditions. These solutions are the only ones under the assumption of constant vorticity

We give necessary and sufficient conditions for the existence of entire solutions bounded or large of the equation \({\mathcal {L}}u - p\psi (u) =0\), where \({\mathcal {L}}\) is either the Laplace operator on \({\mathbb {R}} ^d\), \(d\ge 3\) or the Laplace–Beltrami operator on the harmonic NA group and p is a function whose oscillation tends to zero at infinity at a specified rate. The results apply

This paper studies the boundary behaviour of \(\lambda \)-polyharmonic functions for the simple random walk operator on a regular tree, where \(\lambda \) is complex and \(|\lambda |> \rho \), the \(\ell ^2\)-spectral radius of the random walk. In particular, subject to normalisation by spherical, resp. polyspherical functions, Dirichlet and Riquier problems at infinity are solved, and a non-tangential

In this paper, we study the supercritical problem \((P_\varepsilon )\): \( -\triangle u=K|u|^{4/(n-2)+\varepsilon }u~ \text{ in } \Omega , ~ u=0 \text{ on } \partial \Omega ,\) where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^n\), \(n=3,4\), \(\varepsilon \) is a positive real parameter and K is a \(C^4\) positive function on \({\overline{\Omega }}\). Following the ideas of Bahri et al

We completely classify Levi flat CR structures (that is, CR structures with vanishing Levi form) on three-dimensional real Lie algebras, in terms of their algebraic and almost contact properties.

This paper is devoted to the new shallow-water model (also called the modified Camassa–Holm–Novikov equation) with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the Fokas–Olver–Rosenau–Qiao equation and the Novikov equation as two special cases. It is shown that the Cauchy problem of the modified Camassa–Holm–Novikov equation for the periodic

We deal with a notion of weak binormal and weak principal normal for non-smooth curves of the Euclidean space with finite total curvature and total absolute torsion. By means of piecewise linear methods, we first introduce the analogous notion for polygonal curves, where the polarity property is exploited, and then make use of a density argument. Both our weak binormal and normal are rectifiable curves

A complete classification of left-invariant closed \(G_2\)-structures on Lie groups which are extremally Ricci pinched (i.e., \( {d}\tau = \tfrac{1}{6}|\tau |^2\varphi + \tfrac{1}{6}*(\tau \wedge \tau )\)), up to equivalence and scaling, is obtained. There are five of them, they are defined on five different completely solvable Lie groups and the \(G_2\)-structure is exact in all cases except one,

In this note, we deal with compact generalized (λ, n + m)-Einstein manifolds with nonempty boundary. More precisely, we extend some known results about compact static manifolds for (λ, n + m)-Einstein manifolds, giving topological classifications for its boundary. As a consequence, we obtain some characterizations of the round hemispheres.

If a Hilbert geometry of twice differentiable boundary has two quadratic infinitesimal spheres, then the Hilbert geometry is a Cayley–Klein model of the hyperbolic geometry.

The aim of this paper is to show how the number of conjugacy classes appearing in the product of classes affect the structure of a finite group. The aim of this paper was to show several results about solvability concerning the case in which the power of a conjugacy class is a union of one or two conjugacy classes. Moreover, we show that the above conditions can be determined through the character

We establish a local Lipschitz regularity result of solutions to the Cauchy–Dirichlet problem associated with evolutionary partial differential equations $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {{\,\mathrm{div}\,}}Df(\nabla u) = 0, &{} \quad \text{ in } \, \Omega _T,\\ u=u_0, &{} \quad \text{ on } \, \partial _{{\mathcal {P}}}\, \Omega _T. \end{array} \right. \end{aligned}$$ We

In this paper, we prove a general stability result for higher-order geometric flows on the circle, which basically states that if the initial condition is close to a round circle, the curve evolves smoothly and exponentially fast towards a circle (possibly not the one it started close to), and we improve on known convergence rates (which we believe are almost sharp). The polyharmonic flow is an instance

We construct the tangential k-Cauchy–Fueter complexes on the right quaternionic Heisenberg group, as the quaternionic counterpart of \(\overline{\partial }_b\)-complex on the Heisenberg group in the theory of several complex variables. We can use the \(L^2\) estimate to solve the nonhomogeneous tangential k-Cauchy–Fueter equation under the compatibility condition over this group modulo a lattice. This

We prove continuous dependence on initial data for a backward parabolic operator whose leading coefficients are Osgood continuous in time. This result fills the gap between uniqueness and continuity results obtained so far.

An instanton (E, D) on a (pseudo-)hyperkähler manifold M is a vector bundle E associated with a principal G-bundle with a connection D whose curvature is pointwise invariant under the quaternionic structures of \(T_x M,~x\in M\), and thus satisfies the Yang–Mills equations. Revisiting a construction of solutions, we prove a local bijection between gauge equivalence classes of instantons on M and equivalence

We consider the physical configuration of a container which holds a finite number of movable solid objects and three immiscible fluids. Each fluid volume is prescribed, and the container is completely filled. The configuration is modeled using sets of finite perimeter, and the energy of the configuration is given using the theory of functions of bounded variation. We show that there exists an admissible

Motivated by the large amount of results obtained for minimal and positive constant mean curvature surfaces in several ambient spaces, the aim of this paper is to obtain half-space theorems for properly immersed surfaces in \({\mathbb {R}}^3\) whose mean curvature is given as a prescribed function of its Gauss map. In order to achieve this purpose, we will study the behavior at infinity of a one-parameter

In the present paper, we consider boundary value problems on the real half-line \({\varLambda }:= [0,\infty )\) of the following form $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \Big ({\varPhi }\big (a(t,x(t))\,x'(t)\big )\Big )' = f(t,x(t),x'(t)) \quad \text {a.e.}\,\text {on} \, {\varLambda }, \\ \,\,x(0) = \nu _1,\quad x(\infty ) = \nu _2, \end{array}\right. } \end{aligned}$$ where

We prove the existence of infinitely many sign-changing radial solutions for a p-sub-super critical p-Laplacian Dirichlet problem in a ball. We consider an equation defined by the p-Laplacian operator perturbed by a nonlinearity g(u) that is p-subcritical at \(+\,\infty \) and p-supercritical at \(-\,\infty \). Our results extend those in Castro et al. (Electron. J. Differ. Equ. 2007(111):1–10, 2007)

We give conditions in order to approximate locally uniformly holomorphic covering mappings of the unit ball of \({{\mathbb {C}}^{n}}\) with respect to an arbitrary norm, with entire holomorphic covering mappings. The results rely on a generalization of the Loewner theory for covering mappings which we develop in the paper.

We study the field of moduli of singular K3 surfaces. We discuss both the field of moduli over the CM field and over \({{\mathbb {Q}}}\). As a by-product, we show non-finiteness of isomorphism classes of singular K3 surfaces with field of moduli of bounded degree. Finally, we provide an explicit description of the field of moduli in terms of the 2-torsion of the Galois group of the ring class field

A commutative ring R is said to be endo-Noetherian if the chain of annihilators \({ ann}(a_{1})\subseteq \)\({ ann}(a_{2})\subseteq \cdots \) stabilizes for each sequence \((a_{k})_{k}\) of elements of R (Ndiaye and Gueye in Int J Appl Math 86:871–881, 2013). In this paper, we give equivalent conditions for the power series (resp., polynomial) rings over an Armendariz ring to be endo-Noetherian. We

We prove that if a solution of the time-dependent Schrödinger equation on an homogeneous tree with bounded potential decays fast at two distinct times then the solution is trivial. For the free Schrödinger operator, we use the spectral theory of the Laplacian and complex analysis and obtain a characterization of the initial conditions that lead to a sharp decay at any time. We then adapt the real variable

In this paper, we estimate from above the area of the graph of a singular map u taking a disk to three vectors, the vertices of a triangle, and jumping along three \({\mathcal {C}}^2\)-embedded curves that meet transversely at only one point of the disk. We show that the singular part of the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled

This paper is devoted to a new integrable two-component Novikov equation, lax pairs and bi-Hamiltonian structures. Firstly, the local well-posedness in nonhomogeneous Besov spaces is established by using the Littlewood–Paley theory and transport equations theory. Then, we verify the blow-up that occurs for this system only in the form of breaking waves. Moreover, with analytic initial data, we show

Using quaternionic Feix–Kaledin construction, we provide a local classification of quaternion-Kähler metrics with a rotating \(S^1\)-symmetry with the fixed point set submanifold S of maximal possible dimension. For any real-analytic Kähler manifold S equipped with a line bundle with a real-analytic unitary connection with curvature proportional to the Kähler form, we explicitly construct a holomorphic

A (TE)-structure \(\nabla \) over a complex manifold M is a meromorphic connection defined on a holomorphic vector bundle over \({\mathbb {C}}\times M\), with poles of Poincaré rank one along \(\{ 0 \} \times M\). Under a mild additional condition (the so-called unfolding condition), \(\nabla \) induces a multiplication on TM and a vector field on M (the Euler field), which make M into an F-manifold

Let f be a Lipschitz map from a subset A of a stratified group to a Banach homogeneous group. We show that directional derivatives of f act as homogeneous homomorphisms at density points of A outside a \(\sigma \)-porous set. At all density points of A, we establish a pointwise characterization of differentiability in terms of directional derivatives. These results naturally lead us to an alternate

We state the following weighted Hardy inequality: $$\begin{aligned} c_{o, \mu }\int _{{\mathbb {R}}^N}\frac{\varphi ^2 }{|x|^2}\, \mathrm{d}\mu \le \int _{{\mathbb {R}}^N} |\nabla \varphi |^2 \, \mathrm{d}\mu + K \int _{\mathbb {R}^N}\varphi ^2 \, \mathrm{d}\mu \quad \forall \, \varphi \in H_\mu ^1, \end{aligned}$$ in the context of the study of the Kolmogorov operators: $$\begin{aligned} Lu=\Delta

This note introduces a class of nonlinear Neumann problems on balls expanding with the radii tending toward infinity. Performing singular perturbation arguments, we establish the corresponding concentration phenomenon and refined asymptotic expansions with the precise first two-order terms. In doing so, we obtain the nontrivial boundary structure of solutions with effects coming from the nonlinear

We consider a class of viscous fluids with a general monotone dependence of the viscous stress on the symmetric velocity gradient. We introduce the concept of dissipative solution to the associated initial boundary value problem inspired by the measure-valued solutions for the inviscid (Euler) system. We show the existence as well as the weak–strong uniqueness property in the class of dissipative solutions

In this paper, extending our previous joint work (Hu et al., Math Nachr 291:343–373, 2018), we initiate the study of Hopf hypersurfaces in the homogeneous NK (nearly Kähler) manifold \({\mathbf {S}}^3\times {\mathbf {S}}^3\). First, we show that any Hopf hypersurface of the homogeneous NK \({\mathbf {S}}^3\times {\mathbf {S}}^3\) does not admit two distinct principal curvatures. Then, for the important

The purpose of this article is to study the geometry of static space–times. We show that the energy density and pressure vanish on the boundary of a perfect fluid space–time, provided that it satisfies a suitable condition. Moreover, we provide an upper bound volume growth for geodesic balls of the base of static vacuum space similar to a classical result due to Bishop. In addition, we derive a weak

We define the relative Dolbeault homology of a complex manifold with currents via a Čech approach, and we prove its equivalence with the relative Čech–Dolbeault cohomology as defined by Suwa (in: Singularities–Niigata–Toyama 2007. Advanced studies in pure mathematics, vol 56. Mathematical Society of Japan, Tokyo, pp 321–340, 2009). This definition is then used to compare the relative Dolbeault cohomology

We study the long-time dynamics of the Navier–Stokes equations in the three-dimensional periodic domains with a body force decaying in time. We introduce appropriate systems of decaying functions and corresponding asymptotic expansions in those systems. We prove that if the force has a large-time asymptotic expansion in Gevrey–Sobolev spaces in such a general system, then any Leray–Hopf weak solution

Principal affine open subsets in affine schemes are an important tool in the foundations of algebraic geometry. Given a commutative ring R, R-modules built from the rings of functions on principal affine open subschemes in \({{\,\mathrm{Spec}\,}}R\) using ordinal-indexed filtrations and direct summands are called very flat. The related class of very flat quasi-coherent sheaves over a scheme is intermediate

We show that in the category of preordered sets there is a natural notion of pretorsion theory, in which the partially ordered sets are the torsion-free objects and the sets endowed with an equivalence relation are the torsion objects. Correspondingly, it is possible to construct a stable category factoring out the objects that are both torsion and torsion-free.

We consider a notion of conservation for the heat semigroup associated with a generalized Dirac Laplacian acting on sections of a vector bundle over a noncompact manifold with a (possibly noncompact) boundary under mixed boundary conditions. Assuming that the geometry of the underlying manifold is controlled in a suitable way and imposing uniform lower bounds on the zero-order piece (Weitzenböck potential)

In a measure space \((X,{\mathcal {A}},\mu )\), we consider two measurable functions \(f,g:E\rightarrow {\mathbb {R}}\), for some \(E\in {\mathcal {A}}\). We prove that the property of having equal p-norms when p varies in some infinite set \(P\subseteq [1,+\infty )\) is equivalent to the following condition: $$\begin{aligned} \mu (\{x\in E:|f(x)|>\alpha \})=\mu (\{x\in E:|g(x)|>\alpha \})\quad \text

In this paper, we obtain a vertical–horizontal decomposition formula of Laplacians on manifolds with a special foliation structure. Two Nomizu-type theorems for cohomologies of nilmanifolds follow as applications.

We find a new class of invariant metrics existing on the tangent bundle of any given almost Hermitian manifold. We focus here on the case of Riemannian surfaces, which yield new examples of Kählerian Ricci-flat manifolds in four real dimensions.

We study the asymptotic behavior, as \(\gamma \) tends to infinity, of solutions for the homogeneous Dirichlet problem associated with singular semilinear elliptic equations whose model is $$\begin{aligned} -\Delta u=\frac{f(x)}{u^\gamma }\,\text { in }\Omega , \end{aligned}$$ where \(\Omega \) is an open, bounded subset of \({\mathbb {R}}^{N}\) and f is a bounded function. We deal with the existence